4
$\begingroup$

For 3-dim Poincare Conjecture, the assumption is 'simply connected'. I am wondering whether simply connectedness assumption in 3-dim implies the same homotopy groups as the 3-sphere?

or If we switch the assumption of 'simply connected' to 'homotopy 3-sphere', would it be easier to proof Poincare Conjecuture.

$\endgroup$
5
  • 2
    $\begingroup$ Please do double check your typing: the title of your question is the very first thing people see! $\endgroup$ Sep 23, 2011 at 18:32
  • 3
    $\begingroup$ Yes, in dimension three a closed simply connected 3-manifold is a homotopy sphere. This comes from Poincare duality. $\endgroup$
    – Jim Conant
    Sep 23, 2011 at 18:42
  • $\begingroup$ I know $\pi_2$ and $\pi_3$ can be derived from Poincare duality, but how to derive $\pi_n$ for $n\ge 4$? $\endgroup$
    – user16750
    Sep 23, 2011 at 19:11
  • 5
    $\begingroup$ You look at the map $S^3 \rightarrow M$ generating $\pi_3M = H_3M$ (Hurewicz) and show it induces an isomorphism on cohomology for $\mathbb{Z}$ coefficients. The result follows by a version of Whitehead's theorem. $\endgroup$ Sep 23, 2011 at 19:21
  • $\begingroup$ Despite the editing, the question still asks about the "same homopoy groups" $\endgroup$ Sep 23, 2011 at 20:19

1 Answer 1

10
$\begingroup$

See the fifth paragraph of

http://www.math.cornell.edu/~hatcher/Papers/3Msurvey.pdf

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.